Shaped Post-Field Electronic Oscillations in H

Article pubs.acs.org/JPCA

Shaped Post-Field Electronic Oscillations in H2+ Excited by Two-Cycle Laser Pulses: Three-Dimensional Non-Born−Oppenheimer Simulations Guennaddi K. Paramonov,†,‡ O. Kühn,*,† and André D. Bandrauk¶ †

Institut für Physik, Universität Rostock, Albert-Einstein-Strasse 23-24, D-18059 Rostock, Germany Institut für Chemie, Universität Potsdam, Karl-Liebknecht Strasse 24-25, 14476 Potsdam, Germany ¶ Laboratorie de Chimie Théorique, Faculté des Sciences, Université de Sherbrooke, Sherbrooke, Québec, Canada J1K 2R1 ‡

ABSTRACT: Quantum dynamics of H2+ excited by two-cycle laser pulses with laser carrier frequencies corresponding to the wavelengths λl = 800 and 200 nm (corresponding to the periods τl = 2.667 and 0.667 fs, respectively) and being linearly polarized along the molecular axis have been studied by the numerical solution of the non-Born−Oppenheimer time-dependent Schrödinger equation within a threedimensional (3D) model, including the internuclear distance R and electron coordinates z and ρ. The amplitudes of the pulses have been chosen such that the energies of H2+ after the ends of the laser pulses, ⟨E⟩ ≈ −0.515 au, were close to the dissociation threshold of H2+. It is found that there exists a certain characteristic oscillation frequency ωosc = 0.2278 au (corresponding to the period τosc = 0.667 fs and the wavelength λosc = 200 nm) that plays the role of a “carrier” frequency of temporally shaped oscillations of the expectation values ⟨−∂V/∂z⟩ emerging after the ends of the laser pulses, both at λl = 800 nm and at λl = 200 nm. Moreover, at λl = 200 nm, the expectation value ⟨z⟩ also demonstrates temporally shaped oscillations after the end of the laser pulse. In contrast, at λl = 800 nm, the characteristic oscillation frequency ωosc = 0.2278 au appears as the frequency of small-amplitude oscillations of the slowly varying expectation value ⟨z⟩ which makes, after the end of the pulse, an excursion with an amplitude of about 4.5 au along the z axis and returns back to ⟨z⟩ ≈ 0 afterward. It is found that the period of the temporally shaped post-field oscillations of ⟨−∂V/∂z⟩ and ⟨z⟩, estimated as τshp ≈ 30 fs, correlates with the nuclear motion. It is also shown that vibrational excitation of H2+ is accompanied by the formation of “hot” and “cold” vibrational ensembles along the R degree of freedom. Power spectra related to the electron motion in H2+ calculated for both the laser-driven z and optically passive ρ degrees of freedom in the acceleration form proved to be very interesting. In particular, both odd and even harmonics can be observed.

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INTRODUCTION Nowadays available intense and ultrashort laser pulses allow the study of electron dynamics in the time domain down to the attosecond regime.1−3 Although the single-active valence electron ionization dynamics appears to be reasonably well understood by now,4 recent work has been devoted to multielectron effects, such as electron−hole dynamics in atoms,5 nonsequential ionization,6 or interatomic Coulomb decay.7 Small molecules are prime examples for targets in which the coupling between the laser-excited electronic degrees of freedoms to the nuclear dynamics can be studied by measuring, e.g., the kinetic energy of the ions after Coulomb explosion,8 the photo electrons,9 or employing the recolliding electrons by recording their emitted radiation (high-order harmonic generation)10,11 or their diffraction pattern.12,13 The molecular ion H2+ and the isotopes HD+ and D2+ but also the triangular H32+ are perfectly suited for the study of electron−ion coupling beyond the Born−Oppenheimer approximation in its purest form.14−22 Because of the absence of multielectron effects the interpretation of results is simpler, © XXXX American Chemical Society

and numerical ab initio simulations on a time-dependent Schrödinger equation-level are demanding but possible. The present paper is devoted to the nonlinear response of H2+ to ultrashort (two cycles) and strong, linearly polarized laser pulses of wavelengths λl = 800 nm and λl = 200 nm. The two-cycle laser pulses have been chosen in our study because they provide very fast yet periodic excitation of a molecule. Therefore, both the quantum dynamics of the molecule and especially the related power spectra can be very interesting. The pulse amplitudes are chosen such that the total energies after the laser pulses are ⟨E⟩ ≃ −0.515 au, i.e., just slightly below the dissociation threshold of H2+. The bound electron and nuclear dynamics after the laser pulses are analyzed in detail, including the electron motion in the transversal (ρ) coordinate. For the electronic dipole along the laser polarization direction the same fast electronic post-pulse oscillation frequency ωosc ≃ 0.23 au is Special Issue: Ronnie Kosloff Festschrift Received: November 27, 2015 Revised: January 24, 2016

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DOI: 10.1021/acs.jpca.5b11599 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry A

In eq 2, mn = Mp/2 is the nuclear reduced mass and me = 2MeMp/(Me + 2Mp) is the electron reduced mass. In the atomic units (au) to be used below, we have e = ℏ = Me = 1 and me ≃ 1 and for the field amplitude and intensity , 0 = 5 × 109 V/cm and I0 = 3.5 × 1016 W/cm2, respectively. The Coulomb potential reads

found at both laser wavelengths whereas the modulations due to the nuclear motion differ. The paper is organized as follows: The 3D model of the H2+ and techniques used to solve numerically the time-dependent Schrödinger equation are described in the following section. Afterward, the laser-driven quantum dynamics of H2+ and its free evolution after the ends of the two-cycle laser pulses are presented together with an analysis of the related power spectra. The results are summarized and discussed in the final section.

V (R ,ρ ,z) = e 2(1/R − 1/r1 − 1/r2)

where the electron−proton distances (Figure 1) are

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r1(R ,ρ ,z) = [ρ2 + (z + R /2)2 ]1/2

MODEL, EQUATIONS OF MOTION, AND TECHNIQUES The 3D three-body model with the Coulombic interactions representing the H2+ excited by a laser field linearly polarized along the z axis is shown in Figure 1. The nuclear motion is

r2(R ,ρ ,z) = [ρ2 + (z − R /2)2 ]1/2

(1)

where −e is the electron charge and Mp and Me are the proton and the electron masses, respectively. The homonuclear system H2+ does not have a permanent dipole moment; therefore, its vibrational motion is excited only indirectly due to electronic motion induced by the laser field along the z axis.17,19,20 Electronic motion along the ρ axis occurs due to the wave properties of the electron. The time-dependent Schrödinger equation describing the dynamics of H2+ in the classical laser field ,(t ) reads 1 ∂Ψ ⎞ ℏ2 ⎛ ∂ 2Ψ ∂ ℏ2 ∂ 2Ψ Ψ=− − ⎟ ⎜ 2 + 2 2mn ∂R 2me ⎝ ∂ρ ρ ∂ρ ⎠ ∂t ℏ2 ∂ 2Ψ + V (R ,ρ ,z)Ψ − dz(z) ,(t )Ψ 2me ∂z 2

0 ≤ t ≤ tp

(6)

where , 0 is the amplitude, tp is the pulse duration at the base, and ωl is the laser carrier frequency. Note before proceeding that for a small number of optical cycles per pulse duration, as in the present work for example, the carrier-envelope phase may play a very important role.24 For simplicity we assumed that the carrier-envelope phase in eq 6 is equal to zero, ϕ = 0, in the present study (cf. refs 25 and 26). The numerical technique used to solve the 3D equation of motion, eq 2, has been described in previous works.19,20 The dissociation probability has been calculated with the time- and space-integrated outgoing flux for the nuclear coordinate R; the ionization probabilities have been calculated with the respective fluxes separately for the positive and the negative direction of the z axis as well as for the outer end of the ρ axis. The size of the z-grid has been chosen such as to be substantially larger than the maximum electron excursion α = , 0/ωl 2 . With the maximum electric-field amplitude , 0 = 0.12 au and the minimum laser carrier frequency ωl = 0.056 954 au, used in the present work, the maximum electron excursion is α = 37 au. Specifically, the grid has been adapted such as to accommodate the 3D wave packet in the considered time range. It was damped with the imaginary smooth optical potentials27 starting at |z| > 301 au and at ρ > 271 au for the electronic motion, and at R > 27 au for the nuclear motion. Initially, at t = 0, the H2+ molecule is assumed to be in its ground vibrational and ground electronic state. The wave function of the initial state has been obtained by numerical propagation of the equation of motion in the imaginary time without the laser field (, 0 = 0). This gave a ground state energy of E0 = −0.5791 au, which compares favorably with literature values (see, e.g., ref 28). The amplitudes , 0 of two-cycle laser pulses at λl = 800 and 200 nm used in the present work have been chosen such that the energies of H2+ at the end of the pulses, ⟨E(t=tp)⟩ ≈ −0.515 au, were slightly below the dissociation threshold of H2+, ED = −0.5 au. Note that at λl = 200 nm, the amplitude of the pulse , 0 = 0.03 au proved to be more than twice smaller as compared to that at λl = 800 nm, where , 0 = 0.077 au. It will be instructive to compare the time-dependent expectation values ⟨z⟩ and ⟨ρ⟩ to the respective expectation values ⟨−∂V/∂z⟩ and ⟨−∂V/∂ρ⟩. It is straightforward to show, by making use of Ehrenfest’s theorem, that the acceleration of the expectation value ⟨z⟩ can be written in the following form:

restricted to the polarization direction of the laser electric field, whereas the electron (e) moves in three dimensions with conservation of cylindrical symmetry. Accordingly, two electron coordinates, z and ρ, measured with respect to the center of mass of the two protons (p) are treated explicitly together with the internuclear distance R. The component of the dipole moment of H2+ along the z axis reads23

−

(5)

The time-dependent laser electric field ,(t ) is chosen in the following form:

Figure 1. 3D model of H2+ excited by a laser field linearly polarized along the z axis. The internuclear distance is R, the distances between the electron and each of the two protons are r1 and r2; see eqs 4 and 5.

iℏ

(4)

and

,(t ) = , 0 sin 2(πt /t p) sin(ωlt + ϕ)

dz(z) = −ez[1 + Me /(2M p + Me)]

(3)

(2) B


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Figure 2. Quantum dynamics of H2+ excited by the two-cycle laser pulses at λl = 800 nm (a)−(c) and λl = 200 nm (d)−(f). Parameters of the laser pulses: (a) , 0 = 0.077 au (0.396 GV/cm), ωl = 0.05695 au, tp = 5.33 fs; (d) , 0 = 0.03 au (0.154 GV/cm), ωl = 0.2278 au, tp = 1.33 fs; (b, e) timedependent expectation values ⟨z⟩; (c, f) time-dependent expectation values ⟨ρ⟩.

d2 1 ⟨z⟩ = − [⟨∂V /∂z⟩ + ,(t )] 2 me dt

only during the first optical cycle, t ≤ 2.66 fs. During the second optical cycle and after the end of the two-cycle pulse the expectation value ⟨z⟩ is modulated by small-amplitude oscillations occurring at the frequency ω ≈ 0.2278 au, corresponding to the wavelength of λ ≈ 200 nm. It will be shown below that oscillations with the frequency of ω ≈ 0.2278 au (λ = 200 nm) always exist in the post-pulse free evolution of H2+ related to electronic motion along the z degree of freedom. Due to this reason, the oscillation frequency ω = 0.2278 au will be referred to as the characteristic oscillation frequency and denoted ωosc. With the laser field being aligned along the z axis, the electron degree of freedom ρ is optically passive and excited only due to the wave properties of the electron. It is seen from Figure 2c that at λl = 800 nm, the expectation value ⟨ρ⟩ starts to increase at the end of the first optical cycle, being modulated with (less regular) small-amplitude oscillations taking place at the frequency ωρ ≈ 2ωosc during the second optical cycle and after the end of the two-cycle pulse. The laser-driven dynamics of H2+ excited by the two-cycle laser pulse at λl = 200 nm (ωl = 0.2278 au) is presented in the right panel of Figure 2. It is seen from Figure 2d,e that at a larger carrier frequency, ωl = 0.2278 au, the expectation value ⟨z⟩ does not follow the field even approximately: there is a delay of the electron response to the laser field in the first optical cycle, whereas during the second optical cycle and after the end of the two-cycle pulse the expectation value ⟨z⟩ demonstrates quite regular oscillations taking place at the frequency ω ≈ ωosc, defined above. The time-dependent expectation value ⟨ρ⟩ (Figure 2f) starts to increase at the end of the first optical cycle and demonstrates quite regular oscillations at the frequency ωρ ≈ 2ωosc (τρ ≈ 0.33 fs) during the second optical cycle and after the end of the twocycle pulse. Notice that this difference can be explained in terms of off-resonant polarization of the electron density for λl = 800 nm versus almost resonant driving at λl = 200 nm.

(7)

The time-dependent acceleration d ⟨z⟩/dt will be used in the next section to calculate power spectra generated by the laserdriven electron of H2+ in the acceleration form. Of special interest is the time-dependent acceleration d2⟨ρ⟩/dt2. Because the applied laser field does not excite the ρ degree of freedom directly, its excitation can occur only due to the wave properties of electron driven by the applied laser field along the z axis. Therefore, the electric-field term ,(t ) does not appear in the equation for the acceleration of the expectation value ⟨ρ⟩ at all; i.e., we have 2

2

d2 1 ⟨ρ⟩ = − ⟨∂V /∂ρ⟩ me dt 2

(8)

The time-dependent acceleration d ⟨ρ⟩/dt will be also used in the next section to calculate power spectra related to the optically passive, ρ, degree of freedom in the acceleration form. Although power spectra can be calculated also in the length form, the acceleration form is widely recognized to be often more suitable.29 2

2

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RESULTS Laser-Driven Dynamics and Free Evolution of H2+ on a Short Time Scale and Electron-Field Following. The laser-driven dynamics of H2+ excited by two-cycle laser pulses is presented in Figure 2 by the expectation values ⟨z⟩ and ⟨ρ⟩ at the laser carrier frequency ωl = 0.056 95 au (τl = 2.66 fs, λl = 800 nm, left panel) and at a higher carrier frequency of ωl = 0.2278 au (τl = 0.667 fs, λl = 200 nm, right panel). The respective two-cycle laser pulses are shown in the upper panels (Figure 2a,d). It is seen from Figure 2a,b that at λl = 800 nm, the expectation value ⟨z⟩ follows the field out-of-phase quite well C


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The Journal of Physical Chemistry A As was already mentioned, the time-dependent acceleration d2⟨ρ⟩/dt2 is of special interest because it manifests the wave properties of the electron. The laser-driven dynamics and the subsequent free evolution of H2+ are presented in Figure 3 in

Figure 3. Time-dependent expectation values ⟨−∂V/∂ρ⟩ for H2+ excited along the z axis by two-cycle laser pulses at λl = 800 nm (a) and λl = 200 nm (b). Parameters of the two-cycle laser pulses are as in Figure 2. Figure 4. Snapshots of the reduced probability density P(z,t) in the initial stage. (a) λl = 800 nm; (b) λl = 200 nm. Parameters of the laser pulses are as in Figure 2.

the initial 7 fs for λl = 800 nm and 3 fs for λl = 200 nm with the time-dependent expectation values ⟨−∂V/∂ρ⟩. It is clearly seen from Figure 3 that the electron acceleration d2⟨ρ⟩/dt2 is not equal to zero because the electronic motion along the ρ axis is excited due to the wave properties of the electron. It is also seen from Figures 3 that after the ends of the laser pulses, expectation values ⟨−∂V/∂ρ⟩ demonstrate oscillations at ωρ ≈ 2ωosc. Electronic motion along the laser-driven z degree of freedom is illustrated with the reduced probability density P(z,t) in Figure 4a,b for λl = 800 and 200 nm, respectively. The results in Figure 4a confirm the electron-field out-of-phase following during the first optical cycle of the laser field at λl = 800 nm. Indeed, the maximum of the probability density is slightly shifted to the negative z at t ≈ 1 fs, and strongly shifted to the positive and negative z at t ≈ 2.3 fs and t ≈ 3.5 fs, respectively. In contrast, at λl = 200 nm, Figure 4b, the maximum of the wave packet demonstrates quite large oscillations around ⟨z⟩ already at t > 0.5 fs. Snapshots of the reduced probability density P(z,ρ,t) at several representative times are shown in Figures 5 and 6 for λl = 800 and 200 nm, respectively. In Figure 5 (λl = 800 nm, ωl = 0.056 95 au) the times t = 0.96, 2.29, and 3.47 fs correspond to the turning points of the laser-driven electron motion along the z coordinate (Figure 2b), and the probability densities in the (z, ρ)-plane are distorted accordingly. In contrast, t = 7.09 fs corresponds to ⟨z⟩ ≈ 0 and the probability density in the (z, ρ)plane is almost symmetric. In the case of λl = 200 nm, presented in Figure 6, the electron density is driven along the z axis with four times larger frequency, ωl = 0.2278 au. Representative times t = 1.33 and 1.69 fs correspond to the turning points, whereas t = 1.85 and

Figure 5. Snapshots of the reduced probability density P(z,ρ,t) in the initial stage at λl = 800 nm.

2.19 fs correspond to ⟨z⟩ ≈ 0. It is clearly seen that in all four cases probability densities in the (z, ρ)-plane are substantially distorted, i.e., electronic “ρ-motion” does not follow the laserdriven “z-motion”. It is possible, of course, to find at λl = 200 nm representative times when probability densities are D


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and continues to oscillate around z = 0 with the same frequency ω ≈ ωosc. In contrast, at λl = 200, Figure 7c, the characteristic oscillations at ωosc are by no means small-amplitude. They occur around z = 0 from the very beginning and the characteristic frequency ωosc plays the role of a “carrier” frequency of temporally shaped post-pulse oscillations. The time-dependent expectation values ⟨−∂V/∂z⟩ are presented in Figure 7b,d for λl = 800 nm and λl = 200 nm, respectively. It is clearly seen from Figure 7b,d that in both cases, the post-pulse evolution of the expectation values ⟨−∂V/ ∂z⟩ appears in the form of temporally shaped oscillations with the “carrier” frequency of ωosc. Note that the amplitude of temporally shaped post-pulse oscillations at λl = 800 nm (Figure 7b) is significantly smaller than that occurring at λl = 200 nm (Figure 7d). We emphasize that all post-pulse oscillations, smallamplitude of Figure 7a, small-amplitude temporally shaped of Figure 7b, and temporally shaped large-amplitude (Figure 7c,d), occur at the characteristic frequency ωosc despite very different laser carrier frequencies of the applied laser pulses. Relations of the laser carrier frequencies ωl to the characteristic frequency ωosc are indicated in Figure 7 explicitly. It can be also concluded from Figure 7 that both at λl = 800 nm and at λl = 200 nm, power spectra generated by H2+ due to the laser-induced electron motion along the z coordinate will contain strong peaks at ω ≈ ωosc (see below). Therefore, we can expect that a strong fourth harmonic ωosc ≈ 4 ωl will be generated at λl = 800 and only an “identical” harmonic ωosc ≈ ωl will be generated at λl = 200 nm. The results presented in Figure 7 and discussed above can be rationalized as follows: After the end of the laser pulse, there is a high probability that the H2+ molecule returns back to the initial state ψ0 (in our simulations from 0.65 to 0.75, depending on the laser pulse applied). Then, post-pulse oscillations of the expectation values ⟨z⟩ and ⟨−∂V/∂z⟩, taking place at the characteristic frequency of ωosc, can be qualitatively explained by the assumption that the laser pulse prepares with a certain

Figure 6. Snapshots of the reduced probability density P(z,ρ,t) in the initial stage at λl = 200 nm.

symmetric with respect to an axis perpendicular to z, but they do not in general correspond to ⟨z⟩ ≈ 0. Laser-Driven Dynamics and Free Evolution of H2+ on a Long Time Scale: Temporally Shaped Post-Pulse Oscillations. The time-dependent expectation values ⟨z⟩ and ⟨−∂V/∂z⟩ are presented in Figure 7 on the time scale of 50 fs in the case of λl = 800 nm (left panel) and in the case of λl = 200 nm (right panel). It is seen from Figure 7a that at λl = 800, the electron wave packet makes first a substantial excursion of about 4.5 au in the negative direction of the z axis on the time scale of about 15 fs. During this excursion, the slowly varying time-dependent expectation value ⟨z⟩ is modulated by smallamplitude oscillations occurring at the characteristic frequency ωosc. After the excursion, the electron returns back to ⟨z⟩ ≈ 0

Figure 7. Quantum dynamics of H2+ excited by the two-cycle laser pulses followed by a long-term free evolution, time-dependent expectation values ⟨z⟩ and ⟨−∂V/∂z⟩: (a, b) wavelength λl = 800 nm; (c, d) wavelength λl = 200 nm. Parameters of the laser pulses are as in Figure 2. E


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wavelengths of λl = 800 m (curve a) and 200 nm (curve b). Because the dipole moment of H2+, given by eq 1, does not depend on the internuclear distance R, the nuclear motion of H2+ is activated only by the electronic motion induced by the laser field along the z axis. Indeed, it is easy to see from the comparison of Figures 7 and 8 that there exists a clear correlation between the time-dependent expectation values ⟨z⟩ and ⟨R⟩. At λl = 800 nm, for example, a substantial electron excursion in the negative direction of the z axis (Figure 7a) disturbs the equilibrium configuration of the entire system, resulting in the Coulombic repulsion of the two protons and the elongation of the internuclear distance ⟨R⟩ (Figure 8, curve a). In contrast, at λl = 200 nm, temporally shaped post-pulse oscillations of the electron density take place (Figure 7c,d), and oscillations of the internuclear distance ⟨R⟩ (Figure 8, curve b) nicely correlate with the temporal shaping of ⟨z⟩. It is interesting therefore to find a relation between the frequency of the post-pulse shaping ωshp = 2π/τshp and vibrational frequencies of H2+. It can be concluded from the data presented in Figure 7c,d, and in Figure 8, that the period τshp of post-pulse shaping of electronic motion is τshp ≈ 30 fs, implying the corresponding frequency ωshp ≈ 0.005 au. The frequency of the |v = 0⟩ → |v = 1⟩ vibrational transition in H2+ is ω10 = 0.01 au,28 implying ωshp ≈ ω10/2. It has been shown in our previous works20,32 that vibrational excitation of H2+ at the under-resonant (or below-resonant) laser carrier frequency ωl = ω10/2 is much more efficient as compared to the resonant excitation at ωl = ω10. The physical reason is that prolonged, as compared to the resonant case, time intervals corresponding to the nonequilibrium configuration of the entire system allows the Coulombic repulsion of the two protons to efficiently act on the prolonged time scale, resulting in an efficient vibrational excitation. Taking into account that in the present case shaped post-pulse oscillations of the electron density act with respect to nuclear motion as a periodic external force, we can conclude that the case of ωshp ≈ ω10/2 is very suitable to induce stable and well-correlated electron and nuclear motions. For this reason the electron−nuclei correlation is very well pronounced at λl = 200 nm and less pronounced at λl = 800 nm where the long-range electron excursion in the negative direction of the z axis dominates and therefore the elongation of the internuclear distance ⟨R⟩ is more substantial than at λl = 200 nm. The above given explanation of electron-nuclei correlations was mainly made in terms of expectation values ⟨R⟩ and ⟨z⟩. It can be additionally clarified in terms of electron acceleration ⟨−∂V/∂z⟩ (Figure 7b,d) as follows. At the wavelength of λl = 200 nm, the local maxima of ⟨R⟩ at about 15 and 45 fs (Figure 8, curve b) correspond to the aforementioned 30 fs modulation of proton motion. In Figure 7c, these time moments correspond to local minima of ⟨z⟩, which suggests more symmetric electron distribution because the Coulomb force ⟨−∂V/∂z⟩ presented in Figure 7c is also at local minima. Thus, comparing Figure 8, curve b, to Figure 7c,d shows that at λl = 200 nm the internuclear distance is strongly controlled by expanding− compressing electron motion and electron acceleration along z. This effect is also present weakly at λl = 800 nm (Figure 8, curve a) due to the respective Coulomb force ⟨−∂V/∂z⟩ (Figure 7b) being comparatively small. Note that the time-dependent expectation values ⟨R⟩ presented in Figure 8 provide a very helpful, yet not complete description of vibrational excitation of H2+. Therefore, vibrational dynamics of H2+ is illustrated in more detail in Figure 9, where we show snapshots of the reduced probability

probability a continuum state ψC with the energy EC = E0 + ωosc, where E0 = −0.5971 au is the energy of the initial state ψ0 and EC = −0.3693 au is the energy of the continuum state ψC. In the case of λl = 200 nm (ωl = ωosc = 0.2278 au) the state ψC is prepared with a one-photon transition. In the case of λl = 800 nm (ωl = 0.056 95 au) a four-photon transition is involved. We can also see from Figure 7 that the largest amplitude of temporally shaped post-pulse oscillations takes place at λl = 200 nm (Figure 7d,c) where the one-photon transition ψ0 → ψC is involved. The amplitude of post-pulse oscillations decrease, as can be expected, at λl = 800 nm (Figure 7b) where the fourphoton transition should be involved to prepare the continuum state ψC. Note before proceeding that the importance of vibrational continuum states is well-known. The coupling between vibrational bound and continuum states, for example, affects vibrational transitions among bound states of the H2 molecule in the electronic ground state caused by H atom impact.30 Although the present simulation goes beyond the Born− Oppenheimer approximation a discussion in terms of the Born−Oppenheimer potential curves is instructive. Here, the laser would create a superposition of the electronic ground and excited state. The dipole transition rules requires that there is a node in the electronic wave function; i.e., the state in question would be σ*u 1s. However, in this case the vertical transition frequency would be around 0.4 au, i.e., rather different from ωosc, which points to a more complicated scenario in the presence of non-Born−Oppenheimer effects. The most straightforward way to reveal that the appearance of coherent post-field oscillations is a purely non-Born− Oppenheimer effect is to perform similar numerical simulations at fixed internuclear distances. Our corresponding simulations performed for H2+ revealed that the post-field oscillations do not occur if the Born−Oppenheimer approximation is employed. Similar results have been obtained in our recent work,31 where the post-field muonic oscillations have been found to exist in the non-Born−Oppenheimer treatment of the muonic ddμ and dtμ molecules but did not occur when the Born−Oppenheimer approximation was used. Detailed comparison of the non-Born−Oppenheimer and Born−Oppenheimer quantum dynamics of ddμ and dtμ is presented in our aforementioned work31 (see, in particular, Figures 6 and 7 therein). To reveal the role of nuclear motion, time-dependent expectation values ⟨R⟩ are presented in Figure 8 on the time scale of 50 fs for the cases when H2+ is excited at the

Figure 8. Time-dependent expectation values ⟨R⟩ for H2+ excited by the two-cycle laser pulses at λl = 800 nm (a) and λl = 200 nm (b). Other parameters of the two-cycle laser pulses are as in Figure 2. F


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Figure 10. Time-dependent expectation values ⟨ρ⟩ (a) and ⟨−∂V/∂ρ⟩ (b) for H2+ excited along the z axis by two-cycle laser pulses at λl = 800 nm and λl = 200 nm. Other parameters of the two-cycle laser pulses are as in Figure 2.

group has much less pronounced oscillations with ωρ ≈ 2ωosc and is more similar to the first oscillation at t > 10 fs, which does not have a fine structure of ρ-oscillations at all. In the time interval 10 < t < 50 fs, evolution of ⟨ρ⟩ presents quite irregular oscillations at ωρ ≪ 2ωosc. The time-dependent expectation values ⟨−∂V/∂ρ⟩ are presented in Figure 10b at λl = 800 nm and at λl = 200 nm. After very fast oscillations taking place at ωρ ≈ 2ωosc in the initial stage of 0 < t < 10 fs (see Figure 3a,b for the details), expectation values ⟨−∂V/∂ρ⟩ demonstrate quite a smooth evolution at both wavelengths. For λl = 200 nm we can see a correlation between the electron acceleration d2⟨ρ⟩/dt2 (or ⟨−∂V/∂ρ⟩) and the internuclear distance ⟨R⟩ (Figure 8, curve b). It is seen from the comparison of Figures 8 and 10b that at t ≈ 15 fs both ⟨R⟩ and ⟨−∂V/∂ρ⟩ have their local maxima, subsequently, at t ≈ 30 fs, they reach their local minima, and at t ≈ 45 fs they have local maxima again. From these observations and eq 8, we can conclude that the acceleration d2⟨ρ⟩/dt2 decreases when the internuclear distance ⟨R⟩ decreases (see time interval 15 < t < 30 fs in Figures 10b and 8). In contrast, the acceleration d2⟨ρ⟩/dt2 increases when the internuclear distance ⟨R⟩ increases (see time interval 30 < t < 45 fs in Figures 10b and 8). Therefore, the following hierarchy of coupled electron− nuclear motion can be suggested on a long time scale t > 10 fs (i.e., well after the end of the laser pulse): (i) The laser-driven electron motion along the z axis disturbs the equilibrium initial configuration of H2+ and thus excites the nuclear motion. This electron−nuclei correlation is especially well pronounced at λl = 200 nm when shaped post-pulse oscillations of the electron density along the z axis take place [compare Figure 7 (right panel) to Figure 8, curve b]. (ii) Due to the proton−electron Coulombic attraction, nuclear motion influences electron motion along the ρ axis changing, in particular, electron acceleration d2⟨ρ⟩/dt2. Again, the nuclei−electron correlation is well pronounced at λl = 200 nm, when well correlated oscillations of internuclear distance

Figure 9. Snapshots of the reduced probability density P(R,t) on the long time scale (clipped for visual clarity): (a) λl = 800 nm; (b) λl = 200 nm. Parameters of the laser pulses are as in Figure 2.

density P(R,t) at λl = 800 nm (Figure 9a) and at λl = 200 nm (Figure 9b). Figure 9 reveals that vibrational excitation of H2+ is accompanied by the formation of “hot” and “cold” vibrational ensembles along the R degree of freedom at both λl = 800 nm and λl = 200 nm. The formation of two ensembles of molecules (“hot” and “cold”) was usually observed during multiphoton excitation of more complicated molecules, such as SF6 and CF3I for example.33 It is clearly seen from Figure 9 that “hot” and “cold” ensembles can be formed during the excitation of the simplest molecular ions as well. Finally, in Figure 10 we present the time-dependent expectation values ⟨ρ⟩ and ⟨−∂V/∂ρ⟩ on a long time scale of 50 fs at λl = 800 nm (dashed curves) and λl = 200 nm (solid curves). It is clearly seen from Figure 10a that the timedependent expectation values ⟨ρ⟩ behave quite differently at λl = 800 nm and at λl = 200 nm. For example, at λl = 800 nm the expectation value ⟨ρ⟩ increases to t ≈ 20 fs, being modulated in the very initial stage of 0 < t < 8 fs with small-amplitude oscillations occurring at the frequency of ωρ ≈ 2ωosc (see Figure 2c for more details). Subsequently, at t > 20 fs, expectation value ⟨ρ⟩ smoothly decreases. In contrast, at λl = 200 nm, the time-dependent expectation value ⟨ρ⟩ demonstrates a very different behavior. First of all, the excursion of the electron along the ρ coordinate at λl = 200 nm is much smaller than at λl = 800 nm. A closer look at the λl = 200 nm case in Figure 10a reveals that at the initial stage of 0 < t < 10 fs, the expectation value ⟨ρ⟩ demonstrates three temporally shaped groups of oscillations with the “carrier” frequency ωρ ≈ 2ωosc. Specifically, the first group is presented in Figure 2f in detail; the second group is similar to the first one but the oscillations with ωρ ≈ 2ωosc are slightly less pronounced; and the third G


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The Journal of Physical Chemistry A ⟨R⟩ and electron acceleration d2⟨ρ⟩/dt2 take place (compare Figures 8 and 10b). Power Spectra of Electronic Motion in H2+ Excited by Two-Cycle Laser Pulses. The power spectrum S(ω) of a certain time-dependent expectation value ⟨S(t)⟩ is defined by the squared modulus of the Fourier transform as follows: S(ω) = |

∫0

tf

⟨S(t )⟩ exp( −iωt ) dt |2

We can also see a comparatively low and wide peak in the vicinity of the laser carrier frequency, ω ≈ ωl (ω = 1.17 ωl). Because the ρ degree of freedom of H2+ is not excited directly by the laser field aligned along the z coordinate, spectral features corresponding to the electron motion along the ρ degree of freedom appear only due to the wave properties of electron. In Figure 11b power spectra corresponding to the electron motion along the ρ coordinate are presented. Three dominant peaks in Figure 11b occur at ω ≈ 2 ωl, ω ≈ 4 ωl, and ω ≈ 7 ωl. The strongest peak (at ω = 2.19 ωl) can be assigned to large-amplitude oscillations of ⟨−∂V/∂ρ⟩ in the middle of the 800 nm laser pulse (Figure 3a). The second peak (at ω = 4.43 ωl) corresponds to the dominant peak of the power spectrum generated due to the laser-driven electronic motion along the z coordinate (Figure 11a). The third peak (at ω = 7.31 ωl) appears due to oscillations of ⟨−∂V/∂ρ⟩ at the very end and after the end of the 800 nm laser pulse (Figure 3a). Results obtained for power spectra at λl = 800 nm can be summarized in terms of the characteristic oscillation frequency ωosc as follows. The power spectrum related to the laser-driven z degree of freedom has two dominant peaks: (i) the strongest sharp peak at ω ≈ ωosc and (ii) a lower peak at ω ≈ 0.25 ωosc (i.e., at ω ≈ ωl). The power spectrum related to the optically passive ρ degree of freedom has three dominant peaks: (i) at ω ≈ 0.5 ωosc (the strongest peak); (ii) at ω ≈ ωosc and (iii) at ω ≈ 2 ωosc. Power Spectra at λl = 200 nm. The power spectra Az(ω) and Aρ(ω) are presented in Figure 12a,b, respectively.

(9)

where ⟨S(t )⟩ = ⟨Ψ(t )|S|Ψ(t )⟩

(10)

We shall calculate power spectra in the acceleration form, Az(ω) and Aρ(ω), for the electron coordinates z and ρ, respectively. In that case, the time-dependent expectation value ⟨S(t)⟩ in eq 9 will stand accordingly for d2⟨z⟩/dt2, defined by eq 7, and for d2⟨ρ⟩/dt2 defined by eq 8. In the definitions given above, we take into account as in the previous work34 that the power spectra defined by eqs 9 and 10 do not depend on the sign of S. The upper limit tf of the time-integration in eq 9 has been chosen as follows. For the integrands that do not vanish with time, such as ⟨−∂V/∂ρ⟩ (Figure 10b), we set tf = 50 fs. For the oscillating around zero integrands, such as ⟨−∂V/∂z⟩ (Figure 7b,d), the upper limit tf is chosen, similarly to the previous work,34 close to, but slightly larger than 50 fs, such that the respective integrand is equal to zero at t = tf. In all cases, the power spectra are calculated as functions of frequency ω in units of the respective laser-carrier frequency ωl. Power Spectra at λl = 800 nm. In Figure 11 power spectra Az(ω), generated due to the laser-induced electron motion along the z coordinate, are presented for the case when H2+ is excited at λl = 800 nm. It is clearly seen from Figure 11a that the strongest peak occurs, as expected, at ω ≈ 4 ωl, i.e., at ω ≈ ωosc (ω = 3.95 ωl), indeed, in accordance with the results presented in Figure 7b.

Figure 12. Power spectra generated due to electron motion along the z (a) and ρ (b) coordinate in H2+ excited at λl = 200 nm (ωl = 0.2278 au), , 0 = 0.03 au, and tp = 1.33 fs.

The power spectrum Az(ω) generated due to the laserinduced electron motion along the z coordinate in H2+ (Figure 12a) contains strongly dominant sharp peak at ω ≈ ωl (i.e., at ω ≈ ωosc), as it was expected in accordance with the results presented in Figure 7c,d. The second (lower) peak occurs at ω ≈ 2 ωl. Note that the dominant peak at ω ≈ ωl has a doublet structure. The exact positions of all the three peaks are ω = 0.99 ωl, ω = 1.01 ωl, and ω = 2.03 ωl.

Figure 11. Power spectra in the acceleration form generated due to the laser-driven electronic motion along the z (a) and ρ (b) coordinate in H2+ excited by the two-cycle laser pulse at λl = 800 nm (ωl = 0.056 95 au), , 0 = 0.077 au, and tp = 5.33 fs. H


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molecules,31,35 both odd and even harmonics were generated, with the total number of odd and even harmonics being twice as large as the number of odd harmonics generated in the respective homonuclear isotopes. Note finally that the appearance of even harmonics in Az(w) power spectra of heteronuclear molecules31,35 is a purely nonBorn−Oppenheimer effect, which does not occur if the Born− Oppenheimer approximation is involved.35 Needless to add that the excitation of the ρ degree of freedom by the laser field polarized in a perpendicular direction resulting in the appearance of Aρ(w) power spectra is also a purely nonBorn−Oppenheimer effect.

In contrast, Aρ(ω) (Figure 12b), corresponding to the optically passive ρ degree of freedom is more structured than Az(ω) (Figure 12a). It has, as in the case of Az(ω), dominant maxima in the vicinity of ω ≈ ωl and in the domain of ω ≈ 2 ωl, with the latter being in agreement with the results presented in Figure 3b. Both dominant peaks are accompanied by smaller satellite peaks. The exact positions of all the four peaks are ω = 1.04 ωl, ω = 1.34 ωl, ω = 1.79 ωl, and ω = 2.36 ωl. It is seen that the peak around ω = ωl in Figure 12a is split into two. This may probably be due to the overlap of two components stemming from the two terms on the RHS of eq 7. Taking into account that at λl = 200 nm the laser carrier frequency ωl is equal to the characteristic frequency of the temporally shaped post-pulse oscillations ωosc, we conclude that at λl = 200 nm (similar to λl = 800 nm), oscillations of the optically passive ρ degree of freedom occur both in the domain of ω ≈ ωosc and in the domain of ω ≈ 2 ωosc, with the former frequency domain corresponding to the dominant frequency of the laser-driven “z oscillations”, which occur at ω ≈ ωosc both at λl = 800 nm and at λl = 200 nm. To conclude the section devoted to the power spectra of H2+ excited by two-cycle laser pulses, we would like to emphasize the following features observed. (i) The exact positions of many peaks in power spectra presented in Figures 11 and 12 are not equal to the precise integer multiples of the laser carrier frequency. Harmonics whose frequencies are different from the integer multiplies may be generated by single atoms and molecules as the consequence of resonance effects,35 which definitely takes place in our case of H2+ excited by two-cycle laser pulses. (ii) The power spectra Az(ω) presented in Figures 11a and 12a contain both even and odd harmonics. This is the characteristic feature of two-cycle laser pulses because when more optical cycles are involved in the excitation of a molecule, as in our recent work,31 only odd harmonics related to the z degree of freedom, are generated in the homonuclear muonic ddμ molecule, as suggested by the concept of inversion symmetry,35 whereas both even and odd harmonics are generated in the heteronuclear muonic dtμ molecule due to inversion symmetry breaking. (iii) The power spectrum Aρ(ω) presented in Figure 12b also contains both even and odd harmonics. This is also the characteristic feature of two-cycle laser pulses. Indeed, when more optical cycles are involved, as in our previous work,31 only even harmonics related to the ρ degree of freedom are generated in the homonuclear ddμ molecule, whereas both even and odd harmonics are generated in the heteronuclear dtμ molecule. (iv) The number of generated harmonics correlates with the number of optical cycles involved in the excitation of a molecule. Indeed, in the current case of two-cycle pulses not more than three strong harmonics are generated. In our recent work,31 where similar post-field oscillations were found in muonic molecules ddμ and dtμ, the number of optical cycles at the pulse duration tp = 200 as was 10, and the number of odd harmonics in the power spectrum related to the z degree of freedom of ddμ was 7 (Figure 10a in ref 31). Even harmonics were not generated in the homonuclear ddμ molecule, as suggested by the concept of inversion symmetry.35 When for the excitation of the model H2 molecule35 30 optical cycles were used, the number of generated odd harmonics related to the laser-driven z1 and z2 degrees of freedom was 29. Due to inversion symmetry breaking in the heteronuclear dtμ and HD

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CONCLUSION In the present work, we have studied numerically the quantum dynamics of H2+ excited by linearly polarized along the molecular (z) axis short two-cycle laser pulses with two different carrier frequencies, corresponding to the wavelengths λl = 800 nm and λl = 200 nm. The amplitudes of the pulses have been chosen such that the energy of H2+ at the end of each pulse was close to, but slightly below, the dissociation threshold, specifically, ⟨E⟩ ≈ −0.515 au. The main results obtained in this work can be summarized as follows. (i) There exists a characteristic oscillation frequency ωosc = 0.2278 au (corresponding to the wavelength of λ = 200 nm) that manifests itself as the “carrier” frequency of temporally shaped oscillations of the expectation values ⟨z⟩ and ⟨−∂V/∂z⟩ after the ends of the laser pulses. (ii) Power spectra generated due to the laser-induced electronic motion along the z degree of freedom always have strong and sharp peaks at ω = ωosc which, depending on the laser carrier frequency ωl, can play the role of (a) the higherorder harmonic (ωosc ≈ 4 ωl at λl = 800 nm) and (b) the “identical” harmonic (ωosc ≈ ωl at λl = 200 nm). (iii) Power spectra corresponding to the optically passive ρ degree of freedom of electron always have maxima in the domains of ωρ ≈ ωosc and ωρ ≈ 2 ωosc. In other words, the laser-induced “z oscillations” of electron, which always occur at ω ≈ ωosc, induce the second harmonic of the optically passive “ρ oscillations” and the “identical” harmonic of the “ρ oscillations”. Note that observations (i) and (ii) call for additional investigations at other wavelengths, e.g., at 200 nm < λl < 800 nm and at λl < 200 nm. Preliminary results obtained at λl = 400 nm and at λl = 100 nm confirm the existence of coherent postfield electronic oscillations with the characteristic frequency ωosc = 0.2278 au. Observation (iii) also suggests further investigations. Indeed, calculation of power spectra corresponding to the motion of optically passive ρ degree of freedom at various laser carrier frequencies and various numbers of optical cycles per pulse will provide insight into the fundamental wave properties of electrons. Here, this leads to excitation of one degree of freedom, when another one is directly excited by a periodic external force. This problem has not been systematically addressed yet. We have already mentioned that the previous non-Born−Oppenheimer simulations17,19,20 showed that excitation of the ρ degree of freedom in H2+ and HD+ is not negligible in comparison to that of the z degree of freedom despite polarization of the laser field along the z axis. Recent experimental studies of single ionization of Ar and Ne by circularly polarized laser pulses36 have shown that measured I


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(13) Blaga, C. I.; Xu, J.; DiChiara, A. D.; Sistrunk, E.; Zhang, K.; Agostini, P.; Miller, T. A.; DiMauro, L. F.; Lin, C. D. Imaging Ultrafast Molecular Dynamics With Laser-Induced Electron Diffraction. Nature 2012, 483, 194−197. (14) Chelkowski, S.; Zuo, T.; Atabek, O.; Bandrauk, A. D. Dissociation, Ionization, and Coulomb Explosion of H+2 in an Intense Laser Field by Numerical Integration of the Time-Dependent Schrödinger Equation. Phys. Rev. A: At., Mol., Opt. Phys. 1995, 52, 2977−2983. (15) Chelkowski, S.; Conjusteau, A.; Zuo, T.; Bandrauk, A. D. Dissociative Ionization of H+2 in an Intense Laser Field: ChargeResonance-Enhanced Ionization, Coulomb Explosion, and Harmonic Generation at 600 nm. Phys. Rev. A: At., Mol., Opt. Phys. 1996, 54, 3235−3244. (16) Kawata, I.; Kono, H. Dual Transformation for Wave Packet Dynamics: Application to Coulomb Systems. J. Chem. Phys. 1999, 111, 9498−9508. (17) Kawata, I.; Kono, H.; Fujimura, Y. Adiabatic and Diabatic Responses of H+2 to an Intense Femtosecond Laser Pulse: Dynamics of the Electronic and Nuclear Wave Packet. J. Chem. Phys. 1999, 110, 11152−11165. (18) Kono, H.; Sato, Y.; Tanaka, N.; Kato, T.; Nakai, K.; Koseki, S.; Fujimura, Y. Quantum Mechanical Study of Electronic and Nuclear Dynamics of Molecules in Intense Laser Fields. Chem. Phys. 2004, 304, 203−226. (19) Paramonov, G. K. Ionization and Dissociation of Simple Molecular Ions in Intense Infrared Laser Fields: Quantum Dynamical Simulations for Three-Dimensional Models of HD+ and H+2 . Chem. Phys. Lett. 2005, 411, 350−356. (20) Paramonov, G. K. Vibrational Excitation of Simple Molecular Ions in Resonant and Under-Resonant Strong Laser Fields: Dissociation and Ionization of ppe and pde; Laser-Enhanced Nuclear Fusion in ddμ and dtμ. Chem. Phys. 2007, 338, 329−341. (21) Silva, R. E. F.; Catoire, F.; Rivière, P.; Bachau, H.; Martín, F. Correlated Electron and Nuclear Dynamics in Strong Field Photoionization of H2+. Phys. Rev. Lett. 2013, 110, 113001. (22) Lefebvre, C.; Lu, H. Z.; Chelkowski, S.; Bandrauk, A. D. Electron-Nuclear Dynamics of the One-Electron Nonlinear Polyatomic Molecule H32+ in Ultrashort Intense Laser Pulses. Phys. Rev. A: At., Mol., Opt. Phys. 2014, 89, 023403. (23) Carrington, A.; McNab, I. R.; Montgomerie, C. A. Spectroscopy of the Hydrogen Molecular Ion. J. Phys. B: At., Mol. Opt. Phys. 1989, 22, 3551. (24) Chelkowski, S.; Bandrauk, A. D. Sensitivity of Spatial Photoelectron Distributions to the Absolute Phase of an Ultrashort Intense Laser Pulse. Phys. Rev. A: At., Mol., Opt. Phys. 2002, 65, 061802. (25) Fischer, B.; Kremer, M.; Pfeifer, T.; Feuerstein, B.; Sharma, V.; Thumm, U.; Schröter, C. D.; Moshammer, R.; Ullrich, J. Steering the Electron in H2+ by Nuclear Wave Packet Dynamics. Phys. Rev. Lett. 2010, 105, 223001. (26) Kling, M. F.; Siedschlag, C.; Verhoef, A. J.; Khan, J. I.; Schultze, M.; Uphues, T.; Ni, Y.; Uiberacker, M.; Drescher, M.; Krausz, F.; Vrakking, M. J. J. Control of Electron Localization in Molecular Dissociation. Science 2006, 312, 246. (27) Kaluža, M.; Muckerman, J. T.; Gross, P.; Rabitz, H. Optimally Controlled Five-Laser Infrared Multiphoton Dissociation of HF. J. Chem. Phys. 1994, 100, 4211. (28) Li, H.; Wu, J.; Zhou, B.-L.; Zhu, J.-M.; Yan, Z.-C. Calculations of Energies of the Hydrogen Molecular Ion. Phys. Rev. A: At., Mol., Opt. Phys. 2007, 75, 012504. (29) Burnett, K.; Reed, V.; Knight, J. C. P. Calculation of the Background Emitted During High-harmonic Generation. Phys. Rev. A: At., Mol., Opt. Phys. 1992, 45, 3347. (30) Onda, K. J. Phys. B: At., Mol. Opt. Phys. 1991, 24, 4509. (31) Bandrauk, A. D.; Paramonov, G. K. Excitation of Muonic Molecules ddμ and dtμ by Super-intense Attosecond Soft X-Ray Laser Pulses: Shaped Post-laser-pulse Muonic Oscillations and Enhancement of Nuclear Fusion. Int. J. Mod. Phys. E 2014, 23, 1430014.

lateral expansion of the electron wave packet perpendicular to the laser field is ≈15% larger than predicted by tunneling theory.37 Finally, note that the existence of coherent post-pulse oscillations seems to be a general phenomenon. In a recent work post-pulse oscillations of the induced dipole moment have been obtained in a heavy molecular ion T2+ after its excitation by the UV laser pulse.38 Shaped post-pulse oscillations have also been predicted to occur upon superintense soft X-ray irradiation of muonic molecules.31

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AUTHOR INFORMATION

Corresponding Author

*O. Kühn. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work has been financially supported by the Deutsche Forschungsgemeinschaft through the Sfb 652 (O.K., G.K.P.) which is gratefully acknowledged. A.D.B. thanks the Canada Research Chair and the Humboldt Foundation for the financial support through a Humboldt Research Award. We thank Dieter Bauer (University of Rostock) for stimulating discussions.

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Shaped Post-Field Electronic Oscillations in H

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